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Properties

Label 2-1014-13.9-c1-0-8
Degree $2$
Conductor $1014$
Sign $0.997 + 0.0743i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 3.15·5-s + (−0.499 − 0.866i)6-s + (2.34 + 4.06i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.57 + 2.73i)10-s + (−0.0685 + 0.118i)11-s − 0.999·12-s + 4.69·14-s + (−1.57 + 2.73i)15-s + (−0.5 + 0.866i)16-s + (2.80 + 4.85i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.41·5-s + (−0.204 − 0.353i)6-s + (0.886 + 1.53i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.499 + 0.865i)10-s + (−0.0206 + 0.0357i)11-s − 0.288·12-s + 1.25·14-s + (−0.407 + 0.706i)15-s + (−0.125 + 0.216i)16-s + (0.679 + 1.17i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.997 + 0.0743i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.997 + 0.0743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653631761\)
\(L(\frac12)\) \(\approx\) \(1.653631761\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 3.15T + 5T^{2} \)
7 \( 1 + (-2.34 - 4.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0685 - 0.118i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.80 - 4.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.49 - 4.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.04 + 5.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.425 + 0.736i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.23T + 31T^{2} \)
37 \( 1 + (5.85 - 10.1i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.13 + 3.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.04 - 1.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.98T + 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + (-2.94 - 5.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.35 + 4.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0489 - 0.0847i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.32T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 9.85T + 83T^{2} \)
89 \( 1 + (8.54 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.06 + 1.84i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08756433097173246534154200972, −8.841446215276485183309818325311, −8.279198953114900627842410246592, −7.79974173087126762852370964553, −6.46567903274405846695510839998, −5.49928327831998776092991576802, −4.56566013473004950391989281375, −3.52713404799152067888286014426, −2.57858434348283312330631135127, −1.34269991152639010500229473409, 0.75554664857873964254602497006, 3.10522295488839398001645767557, 3.90681571887339636677644929735, 4.62072950083560231967966791437, 5.30862469852480124425153101137, 7.10481265306983631434671048386, 7.37175443646303220262551836915, 8.017407531324022066965294092574, 8.951138682724143480108724926419, 9.929874104251400817170481590975

Graph of the $Z$-function along the critical line